The propane cylinder is full of mostly liquid propane, with propane vapor in the remainder of the volume. As you use the stove, the vapor is let out to burn, which lowers the pressure of the tank. As a response, some of the liquid propane turns to vapor. In order to make this phase change, the propane absorbs heat from the surroundings (this can be measured as the latent heat of vaporization).
So the tank gets cold because the liquid propane is using heat to vaporize.
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